Abstract
We prove that the second-order unification problem is undecidable even if the functional free variables may only be applied to ground terms. Despite this strong restriction our proof uses elementary techniques, and does not rely on the undecidability of the tenth Hilbert's problem which itself has a very difficult proof. We apply this result to obtain a surprising result of the undecidability of type inference for the Church-style system F - polymorphic λ-calculus.
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